INTRODUCTION TO QUANTUM MATHEMATICS

To complete the concepts of Quantum Computing, the Spin Quantum Tech team wants to organize a trip through the numerical system, to see the Qubits from another point of view

We try to explain mathematical concepts in a simple and fun way so that everyone can enjoy this journey through mathematics, physics, and programming.

Today we will start with Quantum Qubits and the implications they have in daily life

A Qubit from the mathematical point of view

In [1] and [2] we have talked about some interesting facts on the sub-atomic physics related to the quantum computing. In this opportunity we present a Qubit inside the Mathematical Universe: a Qubit is an element of the two dimensional complex (or imaginary) vector space written in the form

where a is a real number, and b is a imaginary one, satisfying

What!!!?, two dimensional? Real number?, Imaginary number?, Imaginary vector space?. Well, in order to be able to enjoy and face this concepts that will take us to the future, in this opportunity we come back to the early beginning of humankind, just to point out that all of us used our fingers, beans or anything to start counting and decide whether mommy gave more sweets to our sister, or our friend scored less points than ourselves, we want to give a glance to the history and talk about the seminal number system of the Natural Numbers.

The natural numbers from inside of us

It is a fact that humankind has obtained marvelous achievements by using the idea of number as we know it nowadays. There are two numbers which appear in our lives at first: one (1) and zero (0), yes! in that precise order. If you want to check just travel inside to your childhood memories, and you find lovely instructions about how to draft a short line or short “stick” and little circle. This constitutes a proof that these two little-giant numbers are present from the beginning of your existence. They are thought as the simplest numbers, but they are even used as the alphabet for the modern and future computation, so they are magically small and the greatest at the same time.

In the pre-history, very certainly, the concept of number appeared to response to the necessity of counting, measuring, and ordering: the plurality of everything around us. Just be aware of your own body, the different kinds of parts, they need to be classified, ordered, measured, and of course counted! As a proof of this continuous process of classification we have lots of rock writings, cave paintings, canvas, among others. Have a look to this picture:

Taken from (3)

The next level we reached to when we were boys or girls is to compute sums, by grouping sets of things from kitchen as beans, glasses, or sets of color pencils, is it true also for you? The professional term for this is arithmetic. Literally, by experimenting with this lovely home laboratories we realized step by step about the properties of the natural numbers set

in our decimal system based on the Arabic symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We start with a unity and the other natural numbers are constructed inductively by adding the unity as many times as we want. Then in history and our own life, a new achievement is gotten: we realized that grouping in sets of then, and positioning numbers one aside the others matters!. For instance, 121 represents more than 112 because

represents a set with one duck, two sets with ten ducks, and one set of one hundred ducks, but

represents a set with two ducks, one set with ten ducks, and one set of one hundred ducks.

The ability of grouping is very important! we can see this in all the most advanced cultures including the current ones, when we use the binary system for computing with the modern machines for example. However, as something curious, it is not apparent the following fact: the human mind is naturally limited to distinguish between numbers greater than four. We invite you to analyze the different ways in which different cultures have represented numbers one to four and number five and so on. Mostly the number five is attributed with a different symbol, see (3).

Historically the zero was not recognized as a number by, mainly, philosophical reasons, since for the ancient mathematicians it was difficult to accept the existence of nothing, of the vacuum, of the nowhere. It appeared not so natural after all if we think of something natural as everything which we can see or feel in our planet or real world. In other words, zero was not a natural number because it fails to count something. On the other hand the number one,  and the arithmetic operations  seem too be important for the barter system or the trading in general.

As we already pointed out, one of the interests of the humans is to compare things, quantities, is to know when something is larger, heavier, longer, higher, or deeper, etc. An usual question is how to decide when two plots of land represent the same amount of space even when they have different geometrical shapes. For sure all the different cultures had to face this problem, however accordingly to Georges Ifrah (1), we can say that in Sumer, an historical region in middle east, south part of the ancient Mesopotamia, between the flood plains of the Euphrates and Tigris rivers, appeared by the first time a metric system based on the natural number 60.

We can define a metric system as a smart way to label each characteristic of a thing with a number, so it is then possible to decide when we have much or less of this characteristic. The first step to construct a metric system is to choose a unity in order to have a basis to compare with, then we can divide this unity in smaller parts in such a way that we can do more precise measurements.

In the centuries XVI and XVII it were discussed the first ideas to create the metric system which we use today. The first legal establishment of a metric system was developed during the French revolution in 1799 when the different systems already existent at that moment had such a bad reputation that it became necessary to replace them by a more standard one. The system then uses the Kilogram, the meter, the second, among other modern unities.

Natural Numbers: some fun, some science,

Formally, in 1889 Giuseppe Peano introduced in his treatise Arithmetices principia, nova method exposits (The principles of arithmetic, presented by a new method; 1889) the standard axiomatization or mathematical definition of the natural numbers, whose main properties are:

• For n, and m natural numbers

(Commutative properties).

• Also

(Associative properties).

• It is true that

(Distributive law).

(Zero is the neutral element).

To show you some interesting facts about the Natural Numbers we recall that, as we learned in school, the multiples of a given natural number k are the elements of the sequence

etc, and we say that m is a divisor of n if and only if n is a multiple of m. So 24 divides 48  because

With this concept we define one of the most important types of numbers: Prime Numbers. The number

is a prime number if and only if its unique divisors are

The first prime numbers are:

Yes!!!! The most beautiful, useful, simplest, misterious, difficult numbers we find with. Here you are with some of its properties:

• They are infinite. This is the Euclid´s Theorem. See (4)

• Every natural number is the product of prime numbers. (Fundamental Theorem of the Arithmetic)

• Every even number, greater than two, is the sum of two primes. (Goldbach´s Conjecture). You can get a million dollars, and immortality if you prove it!!!. Using powerful algorithms, it has been shown to be true for even numbers up to

  

  but we are no sure this will keep being true for all numbers beyond that bound.

• Cryptography is widely supported by prime numbers! So, your money is in hands of this guys. So, we have to be carefully and developed quantum cryptography, one of our main tasks in Spin Quantum Tech.

  There is a mythical history about the Prince of the Mathematics Carl Friedrich Gauss, see (5) to start with a bibliography: it says that when he was about 8-year-old, his teacher asked the class to sum of the first one hundred numbers. Gauss asked before the teacher came out the classroom, 5050! Can you discover how to use the properties the natural numbers we showed you just above to solve this problem???

   

  Coming up next

  As you noticed in your life, it always existed a curiosity for explaining the world around us, and this can be done by means of the numeric universe. This process has been long and has taken thousands of millions of years of work until we can construct and use this tool to create our cellphones, computers, and apply them in sciences, finances, etc. In the next deliveries we will continue to show you the amazing number systems of Integers, rationals, and irrationals, all part of the so-called Real numbers

   

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  Bibliografía

  (1). INTRODUCTION TO QUANTUM COMPUTING (spinqtech.com)

  (2). INTRODUCTION TO QUANTUM COMPUTING PART 2 (spinqtech.com)

  (3). The Universal History of Numbers: from prehistory to the invention of the computer, Georges Ifrah, David Bellos, E. F. Harding, Sophie Wood, Ian Monk, , ISBN: 9780471393405,0471393401 , 2000.  

  (4). An infinite number of primes: proving Euclid's theorem (mathesis-online.com)

  (5).Carl Friedrich Gauss - Wikipedia, la enciclopedia libre

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